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## G = C11×Dic6order 264 = 23·3·11

### Direct product of C11 and Dic6

Aliases: C11×Dic6, C334Q8, C44.3S3, C12.1C22, C132.5C2, C22.13D6, Dic3.C22, C66.18C22, C3⋊(Q8×C11), C4.(S3×C11), C2.3(S3×C22), C6.1(C2×C22), (C11×Dic3).2C2, SmallGroup(264,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C11×Dic6
 Chief series C1 — C3 — C6 — C66 — C11×Dic3 — C11×Dic6
 Lower central C3 — C6 — C11×Dic6
 Upper central C1 — C22 — C44

Generators and relations for C11×Dic6
G = < a,b,c | a11=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×Dic6
Regular action on 264 points
Generators in S264
(1 63 168 38 99 258 172 88 25 20 191)(2 64 157 39 100 259 173 89 26 21 192)(3 65 158 40 101 260 174 90 27 22 181)(4 66 159 41 102 261 175 91 28 23 182)(5 67 160 42 103 262 176 92 29 24 183)(6 68 161 43 104 263 177 93 30 13 184)(7 69 162 44 105 264 178 94 31 14 185)(8 70 163 45 106 253 179 95 32 15 186)(9 71 164 46 107 254 180 96 33 16 187)(10 72 165 47 108 255 169 85 34 17 188)(11 61 166 48 97 256 170 86 35 18 189)(12 62 167 37 98 257 171 87 36 19 190)(49 208 141 249 230 200 226 118 152 76 131)(50 209 142 250 231 201 227 119 153 77 132)(51 210 143 251 232 202 228 120 154 78 121)(52 211 144 252 233 203 217 109 155 79 122)(53 212 133 241 234 204 218 110 156 80 123)(54 213 134 242 235 193 219 111 145 81 124)(55 214 135 243 236 194 220 112 146 82 125)(56 215 136 244 237 195 221 113 147 83 126)(57 216 137 245 238 196 222 114 148 84 127)(58 205 138 246 239 197 223 115 149 73 128)(59 206 139 247 240 198 224 116 150 74 129)(60 207 140 248 229 199 225 117 151 75 130)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204)(205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264)
(1 246 7 252)(2 245 8 251)(3 244 9 250)(4 243 10 249)(5 242 11 248)(6 241 12 247)(13 212 19 206)(14 211 20 205)(15 210 21 216)(16 209 22 215)(17 208 23 214)(18 207 24 213)(25 58 31 52)(26 57 32 51)(27 56 33 50)(28 55 34 49)(29 54 35 60)(30 53 36 59)(37 224 43 218)(38 223 44 217)(39 222 45 228)(40 221 46 227)(41 220 47 226)(42 219 48 225)(61 229 67 235)(62 240 68 234)(63 239 69 233)(64 238 70 232)(65 237 71 231)(66 236 72 230)(73 178 79 172)(74 177 80 171)(75 176 81 170)(76 175 82 169)(77 174 83 180)(78 173 84 179)(85 131 91 125)(86 130 92 124)(87 129 93 123)(88 128 94 122)(89 127 95 121)(90 126 96 132)(97 117 103 111)(98 116 104 110)(99 115 105 109)(100 114 106 120)(101 113 107 119)(102 112 108 118)(133 190 139 184)(134 189 140 183)(135 188 141 182)(136 187 142 181)(137 186 143 192)(138 185 144 191)(145 256 151 262)(146 255 152 261)(147 254 153 260)(148 253 154 259)(149 264 155 258)(150 263 156 257)(157 196 163 202)(158 195 164 201)(159 194 165 200)(160 193 166 199)(161 204 167 198)(162 203 168 197)

G:=sub<Sym(264)| (1,63,168,38,99,258,172,88,25,20,191)(2,64,157,39,100,259,173,89,26,21,192)(3,65,158,40,101,260,174,90,27,22,181)(4,66,159,41,102,261,175,91,28,23,182)(5,67,160,42,103,262,176,92,29,24,183)(6,68,161,43,104,263,177,93,30,13,184)(7,69,162,44,105,264,178,94,31,14,185)(8,70,163,45,106,253,179,95,32,15,186)(9,71,164,46,107,254,180,96,33,16,187)(10,72,165,47,108,255,169,85,34,17,188)(11,61,166,48,97,256,170,86,35,18,189)(12,62,167,37,98,257,171,87,36,19,190)(49,208,141,249,230,200,226,118,152,76,131)(50,209,142,250,231,201,227,119,153,77,132)(51,210,143,251,232,202,228,120,154,78,121)(52,211,144,252,233,203,217,109,155,79,122)(53,212,133,241,234,204,218,110,156,80,123)(54,213,134,242,235,193,219,111,145,81,124)(55,214,135,243,236,194,220,112,146,82,125)(56,215,136,244,237,195,221,113,147,83,126)(57,216,137,245,238,196,222,114,148,84,127)(58,205,138,246,239,197,223,115,149,73,128)(59,206,139,247,240,198,224,116,150,74,129)(60,207,140,248,229,199,225,117,151,75,130), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264), (1,246,7,252)(2,245,8,251)(3,244,9,250)(4,243,10,249)(5,242,11,248)(6,241,12,247)(13,212,19,206)(14,211,20,205)(15,210,21,216)(16,209,22,215)(17,208,23,214)(18,207,24,213)(25,58,31,52)(26,57,32,51)(27,56,33,50)(28,55,34,49)(29,54,35,60)(30,53,36,59)(37,224,43,218)(38,223,44,217)(39,222,45,228)(40,221,46,227)(41,220,47,226)(42,219,48,225)(61,229,67,235)(62,240,68,234)(63,239,69,233)(64,238,70,232)(65,237,71,231)(66,236,72,230)(73,178,79,172)(74,177,80,171)(75,176,81,170)(76,175,82,169)(77,174,83,180)(78,173,84,179)(85,131,91,125)(86,130,92,124)(87,129,93,123)(88,128,94,122)(89,127,95,121)(90,126,96,132)(97,117,103,111)(98,116,104,110)(99,115,105,109)(100,114,106,120)(101,113,107,119)(102,112,108,118)(133,190,139,184)(134,189,140,183)(135,188,141,182)(136,187,142,181)(137,186,143,192)(138,185,144,191)(145,256,151,262)(146,255,152,261)(147,254,153,260)(148,253,154,259)(149,264,155,258)(150,263,156,257)(157,196,163,202)(158,195,164,201)(159,194,165,200)(160,193,166,199)(161,204,167,198)(162,203,168,197)>;

G:=Group( (1,63,168,38,99,258,172,88,25,20,191)(2,64,157,39,100,259,173,89,26,21,192)(3,65,158,40,101,260,174,90,27,22,181)(4,66,159,41,102,261,175,91,28,23,182)(5,67,160,42,103,262,176,92,29,24,183)(6,68,161,43,104,263,177,93,30,13,184)(7,69,162,44,105,264,178,94,31,14,185)(8,70,163,45,106,253,179,95,32,15,186)(9,71,164,46,107,254,180,96,33,16,187)(10,72,165,47,108,255,169,85,34,17,188)(11,61,166,48,97,256,170,86,35,18,189)(12,62,167,37,98,257,171,87,36,19,190)(49,208,141,249,230,200,226,118,152,76,131)(50,209,142,250,231,201,227,119,153,77,132)(51,210,143,251,232,202,228,120,154,78,121)(52,211,144,252,233,203,217,109,155,79,122)(53,212,133,241,234,204,218,110,156,80,123)(54,213,134,242,235,193,219,111,145,81,124)(55,214,135,243,236,194,220,112,146,82,125)(56,215,136,244,237,195,221,113,147,83,126)(57,216,137,245,238,196,222,114,148,84,127)(58,205,138,246,239,197,223,115,149,73,128)(59,206,139,247,240,198,224,116,150,74,129)(60,207,140,248,229,199,225,117,151,75,130), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264), (1,246,7,252)(2,245,8,251)(3,244,9,250)(4,243,10,249)(5,242,11,248)(6,241,12,247)(13,212,19,206)(14,211,20,205)(15,210,21,216)(16,209,22,215)(17,208,23,214)(18,207,24,213)(25,58,31,52)(26,57,32,51)(27,56,33,50)(28,55,34,49)(29,54,35,60)(30,53,36,59)(37,224,43,218)(38,223,44,217)(39,222,45,228)(40,221,46,227)(41,220,47,226)(42,219,48,225)(61,229,67,235)(62,240,68,234)(63,239,69,233)(64,238,70,232)(65,237,71,231)(66,236,72,230)(73,178,79,172)(74,177,80,171)(75,176,81,170)(76,175,82,169)(77,174,83,180)(78,173,84,179)(85,131,91,125)(86,130,92,124)(87,129,93,123)(88,128,94,122)(89,127,95,121)(90,126,96,132)(97,117,103,111)(98,116,104,110)(99,115,105,109)(100,114,106,120)(101,113,107,119)(102,112,108,118)(133,190,139,184)(134,189,140,183)(135,188,141,182)(136,187,142,181)(137,186,143,192)(138,185,144,191)(145,256,151,262)(146,255,152,261)(147,254,153,260)(148,253,154,259)(149,264,155,258)(150,263,156,257)(157,196,163,202)(158,195,164,201)(159,194,165,200)(160,193,166,199)(161,204,167,198)(162,203,168,197) );

G=PermutationGroup([(1,63,168,38,99,258,172,88,25,20,191),(2,64,157,39,100,259,173,89,26,21,192),(3,65,158,40,101,260,174,90,27,22,181),(4,66,159,41,102,261,175,91,28,23,182),(5,67,160,42,103,262,176,92,29,24,183),(6,68,161,43,104,263,177,93,30,13,184),(7,69,162,44,105,264,178,94,31,14,185),(8,70,163,45,106,253,179,95,32,15,186),(9,71,164,46,107,254,180,96,33,16,187),(10,72,165,47,108,255,169,85,34,17,188),(11,61,166,48,97,256,170,86,35,18,189),(12,62,167,37,98,257,171,87,36,19,190),(49,208,141,249,230,200,226,118,152,76,131),(50,209,142,250,231,201,227,119,153,77,132),(51,210,143,251,232,202,228,120,154,78,121),(52,211,144,252,233,203,217,109,155,79,122),(53,212,133,241,234,204,218,110,156,80,123),(54,213,134,242,235,193,219,111,145,81,124),(55,214,135,243,236,194,220,112,146,82,125),(56,215,136,244,237,195,221,113,147,83,126),(57,216,137,245,238,196,222,114,148,84,127),(58,205,138,246,239,197,223,115,149,73,128),(59,206,139,247,240,198,224,116,150,74,129),(60,207,140,248,229,199,225,117,151,75,130)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204),(205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264)], [(1,246,7,252),(2,245,8,251),(3,244,9,250),(4,243,10,249),(5,242,11,248),(6,241,12,247),(13,212,19,206),(14,211,20,205),(15,210,21,216),(16,209,22,215),(17,208,23,214),(18,207,24,213),(25,58,31,52),(26,57,32,51),(27,56,33,50),(28,55,34,49),(29,54,35,60),(30,53,36,59),(37,224,43,218),(38,223,44,217),(39,222,45,228),(40,221,46,227),(41,220,47,226),(42,219,48,225),(61,229,67,235),(62,240,68,234),(63,239,69,233),(64,238,70,232),(65,237,71,231),(66,236,72,230),(73,178,79,172),(74,177,80,171),(75,176,81,170),(76,175,82,169),(77,174,83,180),(78,173,84,179),(85,131,91,125),(86,130,92,124),(87,129,93,123),(88,128,94,122),(89,127,95,121),(90,126,96,132),(97,117,103,111),(98,116,104,110),(99,115,105,109),(100,114,106,120),(101,113,107,119),(102,112,108,118),(133,190,139,184),(134,189,140,183),(135,188,141,182),(136,187,142,181),(137,186,143,192),(138,185,144,191),(145,256,151,262),(146,255,152,261),(147,254,153,260),(148,253,154,259),(149,264,155,258),(150,263,156,257),(157,196,163,202),(158,195,164,201),(159,194,165,200),(160,193,166,199),(161,204,167,198),(162,203,168,197)])

99 conjugacy classes

 class 1 2 3 4A 4B 4C 6 11A ··· 11J 12A 12B 22A ··· 22J 33A ··· 33J 44A ··· 44J 44K ··· 44AD 66A ··· 66J 132A ··· 132T order 1 2 3 4 4 4 6 11 ··· 11 12 12 22 ··· 22 33 ··· 33 44 ··· 44 44 ··· 44 66 ··· 66 132 ··· 132 size 1 1 2 2 6 6 2 1 ··· 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C11 C22 C22 S3 Q8 D6 Dic6 S3×C11 Q8×C11 S3×C22 C11×Dic6 kernel C11×Dic6 C11×Dic3 C132 Dic6 Dic3 C12 C44 C33 C22 C11 C4 C3 C2 C1 # reps 1 2 1 10 20 10 1 1 1 2 10 10 10 20

Matrix representation of C11×Dic6 in GL2(𝔽23) generated by

 12 0 0 12
,
 0 9 5 16
,
 15 17 7 8
G:=sub<GL(2,GF(23))| [12,0,0,12],[0,5,9,16],[15,7,17,8] >;

C11×Dic6 in GAP, Magma, Sage, TeX

C_{11}\times {\rm Dic}_6
% in TeX

G:=Group("C11xDic6");
// GroupNames label

G:=SmallGroup(264,18);
// by ID

G=gap.SmallGroup(264,18);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-3,220,461,226,4404]);
// Polycyclic

G:=Group<a,b,c|a^11=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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